Proportional-integral-derivative control systems (P-I-D) are well known in the control system art, and are employed where precise control of a process or a condition of a machine is required. P-I-D control systems are designed to overcome permanent offsets which are common to simple proportional control systems, and to overcome the slow response times and overshoots found in the more complex proportional-integral control systems.
In simple control systems the error between the actual condition and the desired condition of a process or machine being controlled is used to directly drive a controlling element. The controlling element then urges the condition of the process or machine toward the desired state. In certain applications, especially in control systems controlling air or electrically driven elements, some amount of force must be left on the actuator just to counter resistive forces in the driven elements. This is true, for example, where air pressure is needed in a piston to support a weight, or an electric current is needed to maintain temperature in a process that would normally cool off. This is less true in a hydraulic cylinder, where the controlled item is an amount of high pressure liquid rather than the pressure of a compressible gas. Still, such systems can inherently include a "droop" in the condition being controlled that is somewhat proportional to the resisting force. This droop takes the form of a variable offset which may cause intolerable inaccuracies in the system.
Proportional-integral systems are employed to counter the variable offset problem of simple proportional systems. In such systems an integral term is added to the controlling signal determination. The integral term is the error compounded over time and then proportioned by a gain term. The integral term is added to the proportional signal to bring the machine or process closer to the desired condition or setpoint in an iterative manner. If an error still exists after the integral term is updated, the remaining error will again be integrated with the previous integral correction and again added to the proportional signal, thus bringing the condition a bit closer to the desired state.
As the controlled condition gets closer to the desired state or setpoint, the amount by which the integral term increases will get smaller, theoretically approaching zero as the error goes to zero. The integral signal will then remain at its current value and thereby hold the controlled device at its desired condition or setpoint. If the device being controlled overshoots the desired condition, the new error becomes negative and operates to reduce the magnitude of the integral term. As such, the effect of the integral term on the controlled device is reduced, which brings the controlled device back toward the desired condition or setpoint.
As mentioned above, among the problems of proportional-integral systems is slow response time and overshoots. The nature of an integration operation dictates that the impact of the integral term builds slowly as the sum of the existing error is built up. The rate of build-up can be increased by multiplying the integral by an appropriate gain term. However, if the gain term is set too high, overshoots can become a problem. Further, if the condition being controlled has an inertial component, an overshoot will be present.
The problem of overshoots and response time is addressed by P-I-D control systems, wherein a time derivative term is included to increase the initial response to an increasing error term and to reduce the rate at which the condition being controlled approaches the desired condition. The time derivative term, also known as the rate signal, is the derivative of the error with respect to time; i.e., the rate of change of the error. Thus, the derivative term is large when the amount of change is large, and small when the amount of change is small.
The time derivative term is introduced as a negative term in the controlling signal to slow down the rate at which the error is driven to zero. If the error is being reduced rapidly, the time derivative term will act to reduce the controlling signal and will slow down the rate at which the process approaches the setpoint. If the error term is constant, the derivative term is zero.
If the device being controlled is at the setpoint and is then rapidly driven off the setpoint by an external force, or the setpoint changes rapidly, the time derivative term will be immediately nonzero and start to urge the device being controlled back to the desired condition. This will happen before the error term alone which is present in the proportional signal term is large enough to significantly oppose the condition, and before the integral signal has had enough time to build up to increase the restoring signal.
When the controlling signal has built up enough to counter the offsetting force, the rate signal will, of course, have gone to zero. However, at that point, the integral signal will be building up at the maximum rate (since there is maximum error). The error will start to return towards zero. The time derivative of this return towards zero causes a non-zero rate signal to be generated which has an opposite polarity from when the error was increasing. This slows down the return to zero to prevent overshoot.
In a P-I-D system, then, the error of the system is increased by an integral signal which reflects the cumulative error over time, and decreased by a derivative signal which reflects the rate that the error is approaching zero. The integral signal prevents a permanent offset with the system never reaching zero error, while the derivative signal reduces the probability of overshoot. As such, a three element control system of the prior art can be modelled as follows: EQU Output=C.sub.1 (C.sub.2 E+C.sub.3 .intg.Edt-C.sub.4 (dE/dt))
Where,
Output=the signal that will go to the controlling element, PA1 C.sub.1 =a constant for increasing or decreasing the magnitude of the Output, PA1 E=the error term: the numerical difference between the desired condition (or order signal) and the actual condition (or feedback signal), PA1 C.sub.2, C.sub.3, and C.sub.4 =proportionalizing constants for the proportional term, the integral term, and the derivative term, respectively. PA1 O=order or reference signal, PA1 F=sensed parameter signal or feedback signal, PA1 E.sub.-1 =previous error signal, ##EQU2## and where N=number of previous cycles, PA1 CS=controlling signal, PA1 C.sub.1 =constant of proportionality for controlling signal, PA1 C.sub.2 =proportional component sizing factor, PA1 C.sub.3 =integral component sizing factor, PA1 C.sub.4 =derivative component sizing factor, PA1 C.sub.5 =error sizing factor to the integral component, PA1 C.sub.6 =order signal offset addend, PA1 C.sub.7 =sensed parameter signal offset addend, PA1 C.sub.8 =order signal sizing factor, PA1 C.sub.9 =sensed parameter signal sizing factor, PA1 C.sub.11 =maximum controlling signal level, and PA1 C.sub.12 =previous integral component sizing factor.
This form of P-I-D control system has been used to control positional equipment and other types of systems for many years. Reference is made to U.S. Pat. No. 4,430,698 to Harris in which such a P-I-D system is described in greater detail.
Until recently P-I-D control systems have been implemented in analog form. The article by J. Fishbeck, "Writing P-I-D Control Loops Easily in BASIC", Control Engineer, October, 1978, pp. 45-47, shows that P-I-D control systems can be implemented in digital form using a digital computer.
However, despite the recent advances in P-I-D control systems, such systems are typically custom designed for a particular application, and thus not resorted to in applications where cost and flexibility are of concern. Typically, current P-I-D control systems cannot be modified easily from one application to another. Thus, where a new controlling element being driven by the control system requires a different drive signal range or a particular offset, current P-I-D control systems may not be able to accommodate such requirements. Moreover, typical P-I-D control systems are designed around the sensor benng used to monitor the condition of the process or machine being controlled. As such, if it is desired that a different sensor be utilized, a substantial redesign of the system may be required.
Further, present P-I-D control systems continue to have inaccuracies which are not easily remedied with current P-I-D architectures.
Moreover, present P-I-D systems cannot be easily adapted to changing conditions of the process or machine being controlled.